toqito.state_metrics.fidelity
=============================

.. py:module:: toqito.state_metrics.fidelity

.. autoapi-nested-parse::

   Fidelity is a metric that qualifies how close two quantum states are.





Module Contents
---------------

.. py:function:: fidelity(rho, sigma)

   Compute the fidelity of two density matrices [@WikiFidQuant].

   Calculate the fidelity between the two density matrices `rho` and `sigma`, defined by:

   \[
       ||\sqrt(\rho) \sqrt(\sigma)||_1,
   \]

   where \(|| \cdot ||_1\) denotes the trace norm. The return is a value between \(0\) and \(1\), with
   \(0\) corresponding to matrices `rho` and `sigma` with orthogonal support, and \(1\)
   corresponding to the case `rho = sigma`.

   .. rubric:: Examples

   Consider the following Bell state

   \[
       u = \frac{1}{\sqrt{2}} \left( |00 \rangle + |11 \rangle \right) \in \mathcal{X}.
   \]

   The corresponding density matrix of \(u\) may be calculated by:

   \[
       \rho = u u^* = \frac{1}{2} \begin{pmatrix}
                        1 & 0 & 0 & 1 \\
                        0 & 0 & 0 & 0 \\
                        0 & 0 & 0 & 0 \\
                        1 & 0 & 0 & 1
                      \end{pmatrix} \in \text{D}(\mathcal{X}).
   \]

   In the event where we calculate the fidelity between states that are identical, we should obtain the value of
   \(1\). This can be observed in `|toqito⟩` as follows.

   ```python exec="1" source="above"
   import numpy as np
   from toqito.state_metrics import fidelity

   rho = 1 / 2 * np.array(
       [[1, 0, 0, 1],
        [0, 0, 0, 0],
        [0, 0, 0, 0],
        [1, 0, 0, 1]]
   )
   sigma = rho

   print(fidelity(rho, sigma))
   ```

   :raises ValueError: If matrices are not density operators.

   :param rho: Density operator.
   :param sigma: Density operator.

   :returns: The fidelity between `rho` and `sigma`.